For this word, the 便 comes from 便所, which you know as "toilet." A toilet secret is one you take with you to the bathroom, one that you don't want people to know about. This is constipation.

This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own. The ひ changes to ぴ because "P" stands for "Poop."